DESCRIPTIVE SET THEORY
DESCRIPTIVE SET THEORY
YIANNIS N. MOSCHOVAKIS Professor of Mathematics University of California, Los Angeles and Emeritus Professor of Mathematics University of Athens, Athens, Greece
[email protected] April 8, 2009
This book is dedicated to the memory of my father Nicholas a good and gentle man
CONTENTS
Preface to the second edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Preface to the first edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
About this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Chapter 1. The basic classical notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1A. Perfect Polish spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1B. The Borel pointclasses of finite order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1C. Computing with relations; closure properties . . . . . . . . . . . . . . . . . . . . . . . 1D. Parametrization and hierarchy theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1E. The projective sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1F. Countable operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1G. Borel functions and isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1H. Historical and other remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 9 13 18 26 29 33 37 46
Chapter 2. κ-Suslin and ë-Borel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2A. The Cantor-Bendixson Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2B. κ-Suslin sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2C. Trees and the Perfect Set Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2D. Wellfounded trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2E. The Suslin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2F. Inductive analysis of projections of trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2G. The Kunen-Martin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2H. Category and measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2I. Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49 50 51 57 62 65 70 74 79 85
Chapter 3. Basic notions of the effective theory . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3A. Recursive functions on the integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3B. Recursive presentations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3C. Semirecursive pointsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3D. Recursive and Γ-recursive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3E. The Kleene pointclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3F. Universal sets for the Kleene pointclasses. . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3G. Partial functions and the substitution property . . . . . . . . . . . . . . . . . . . . . 130 3H. Codings, uniformity and good parametrizations . . . . . . . . . . . . . . . . . . . . 135 3I. Effective theory on arbitrary (perfect) Polish spaces . . . . . . . . . . . . . . . . . 141 vii
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CONTENTS 3J.
Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Chapter 4. Structure theory for pointclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4A. The basic representation theorem for Π11 sets . . . . . . . . . . . . . . . . . . . . . . . 145 4B. The prewellordering property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4C. Spector pointclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4D. The parametrization theorem for ∆ ∩ X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4E. The uniformization theorem for Π11 , Σ12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4F. Additional results about Π11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4G. Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Chapter 5. The constructible universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5A. Descriptive set theory in L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5B. Independence results obtained by the method of forcing . . . . . . . . . . . . . 214 5C. Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Chapter 6. The playful universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6A. Infinite games of perfect information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6B. The First Periodicity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 6C. The Second Periodicity Theorem; uniformization . . . . . . . . . . . . . . . . . . . 235 6D. The game quantifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6E. The Third Periodicity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 6F. The determinacy of Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 6G. Measurable cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 6H. Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 G
Chapter 7. The recursion theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 7A. Recursion in a Σ∗ -pointclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 7B. The Suslin-Kleene Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 7C. Inductive definability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 7D. The completely playful universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 7E. Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 7F. Results which depend on the Axiom of Choice . . . . . . . . . . . . . . . . . . . . . . 341 Chapter 8. Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 8A. Structures and languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 8B. Elementary definability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 8C. Definability in the universe of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 ¨ 8D. Godel’s universe of constructible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 8E. Absoluteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 8F. The basic facts about L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 8G. Regularity results and inner models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 8H. On the theory of indiscernibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 8I. Some remarks about strong hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 8J. Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 The axiomatics of pointclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
PREFACE TO THE SECOND EDITION
There was no question of “updating” this book nearly thirty years after it was first published—in 1980, volume 100 in the Studies in Logic series of North Holland. The only completely rewritten sections are 6F, which gives a proof of the determinacy of Borel sets (a version of Martin’s second proof not available in 1980) and 7F, where the question of how much choice is needed (especially) to prove Borel determinacy is examined. There is also a new, brief section 3I on the relativization method of proof, which has baffled some of the not-so-logically minded readers. Beyond that, the main improvements over the first edition are that - this one has many fewer errors (I hope); - the bibliography has been completed and expanded with a small selection of relevant, more recent publications; - and many passages have been rewritten. (It has been said that the most basic instinct in man is not for food or sex but to edit someone else’s writing—and the urge to edit one’s own writing is, apparently, even stronger.) There have been two major developments in Descriptive Set Theory since 1980 which have fundamentally changed the subject. One is the establishment of a robust connection between determinacy hypotheses, large cardinal axioms and inner model theory, starting with Martin and Steel [1988] and Woodin [1988], to such an extent that one cannot now understand any of these parts of set theory without also understanding the others. I have added some “forward references” to these developments when they touch on questions that were formulated in the book. The other is the explosion in applications of Descriptive Set Theory to other parts of mathematics, cf. Kechris [1995]. This area really took off with Harrington, Kechris, and Louveau [1990] which (with the work that followed it) established the study of definable equivalence relations on Polish spaces as a subject of its own, with deep connections to classical mathematics. It was not possible to point to this work in this revision, especially as the basic result in Silver [1980] was not (for some reason) included in the original. Many of the notions and techniques introduced in this book have been used heavily in these developments, notably scales and the application of effective methods to the “classical” theory. Some of it has become obsolete, of course; but I do not believe that its self-contained, foundationally motivated and unified introduction to the effective theory and the consequences of determinacy hypotheses has been duplicated. I am grateful to all those who have sent me comments and corrections, including (from the incomplete records that I have) Ben Miller, Mike Brady, Vassilis Gregoriades, ix
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Preface to the second edition
Tonny Hurkens, Aleko Kechris, Tony Martin, Itay Neeman, Richard Shore and John Steel. I am especially grateful to Christos Kapoutsis who set the manuscript in beautiful LATEX several years ago—and I apologize to him that it took me so long to do my part and finish the job. Paleo Faliro, Greece
July 29, 2008
PREFACE TO THE FIRST EDITION
This book was conceived in the winter of 1970 when I heard that I was getting a Sloan Fellowship and I thought I would take a year off to write a book. It took a bit longer than that, but I have many good excuses. I am grateful to the Sloan Foundation, the National Science Foundation and the University of California for their financial support—and to the Mathematics Department at UCLA for the stimulating and pleasant working environment that it provides. One often sees in prefaces long lists of persons who have contributed to the project in one way or another and I hope I will be forgiven for not complying with tradition; in my case any reasonably complete list would have to start with Lebesgue and increase the size of the book beyond the publisher’s indulgence. I will, however, mention my student Chris Freiling who read carefully through the entire final version of the manuscript and corrected all my errors. My wife Joan is the only person who really knows how much I owe to her and she is too kind to tell. But I know too. Finally, my deepest feelings of gratitude and appreciation are reserved for the very few friends with whom I have spent so many hours during the last ten years arguing about descriptive set theory; Bob Solovay and Tony Martin in the beginning, Aleko Kechris, Ken Kunen and Leo Harrington a little later. Their influence on my work will be obvious to anyone who glances through this book and I consider them my teachers—although of course, they are all so much younger than me. No doubt I would still work in this field if they were all priests or generals—but I would not enjoy it half as much. Santa Monica, California
December 22, 1978
Added in proof. I am deeply grateful to Dr. Haimanti Sarbadhikari who read the first seven chapters in proof and corrected all the errors missed by Chris Freiling. I am also indebted to Anna and Nicholas for their substantial help in constructing the indexes and to Tony Martin for the sustenance he offered me during the last stages of this work. xi
ABOUT THIS BOOK
My aim in this monograph is to give a brief but coherent exposition of the main results and methods of descriptive set theory. I have made no attempt to be complete; in a subject so broad this would degenerate into a long catalog of specialized results which would cover up the main thread. On the contrary, I have tried very hard to be selective, so that the central ideas stand out. Much of the material is in the exercises. A very few of them are simple, to test the reader’s comprehension, and a few more give interesting extensions of the theory or sidelines. The vast majority of the exercises are an integral part of the monograph and would be normally billed “theorems.” There are extensive “hints” for them, proofs really, with some of the details omitted. I have tried hard to attribute all the important results and ideas to those who invented them but this was not an easy task and I have undoubtedly made many errors. There is no suggestion that unattributed results are mine or are published here for the first time. When I do not give credit for something, the most likely explanation is that I could not determine the correct credit. My own results are immodestly attributed to me, including those which are first published here. Many of the references are in the historical sections at the end of each chapter. The paragraphs of these sections are numbered and the footnotes in the body of the text refer to these paragraphs—each time meaning the section at the end of the chapter where the reference occurs. In a first reading, it is best to skip these historical notes and read them later, after one is familiar with the material in the chapter. The order of exposition follows roughly the historical development of the subject, simply because this seemed the best way to do it. It goes without saying that the classical results are presented from a modern point of view and using modern notation. What appeals to me most about descriptive set theory is that to study it you must really understand so many things: you need a little bit of topology, analysis and logic, a good deal of recursive function theory and a great deal of set theory, including constructibility, forcing, large cardinals and determinacy. What makes the writing of a book on the subject so difficult is that you must explain so many things: a little bit of topology, analysis and logic, a good deal of recursive function theory, etc. Of course, one could aim the book at those who already know all the prerequisites, but chances are that these few potential readers already know descriptive set theory. My aim has been to make this material accessible to a mathematician whose particular field of specialization could be anything, but who has an interest in set theory, or at least what used to be called “the theory of pointsets.” He certainly knows whatever little topology and analysis are required, because he learned that as an undergraduate, and he has read Halmos’ Naive Set Theory [1960] or a similar text. Beyond that, what he needs to
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About this book
read this book is patience and a basic interest in the central problem of descriptive set theory and definability theory in general: to find and study the characteristic properties of definable objects.
INTRODUCTION
The roots of Descriptive Set Theory go back to the work of Borel, Baire and Lebesgue around the turn of the 20th century, when the young French analysts were trying to come to grips with the abstract notion of a function introduced by Dirichlet and Riemann. A function was to be an arbitrary correspondence between objects, with no regard for any method or procedure by which this correspondence could be established. They had some doubts whether so general a concept should be accepted; in any case, it was obvious that all the specific functions which were studied in practice were determined by simple analytic expressions, explicit formulas, infinite series and the like. The problem was to delineate the functions which could be defined by such accepted methods and search for their characteristic properties, presumably nice properties not shared by all functions. Baire was first to introduce in his Thesis [1899] what we now call Baire functions (of several real variables), the smallest set which contains all continuous functions and is closed under the taking of (pointwise) limits. He gave an inductive definition: the continuous functions are of class 0 and for each countable ordinal î, a function is of class î if it is the limit of a sequence of functions of smaller classes and is not itself of lower class. Baire, however, concentrated on a detailed study of the functions of class 1 and 2 and he said little about the general notion beyond the definition. The first systematic study of definable functions was Lebesgue’s [1905], Sur les fonctions repr´esentables analytiquement. This beautiful and seminal paper truly started the subject of descriptive set theory. Lebesgue defined the collection of analytically representable functions as the smallest set which contains all constants and projections (x1 , x2 , . . . , xn ) 7→ xi and which is closed under sums, products and the taking of limits. It is easy to verify that these are precisely the Baire functions. Lebesgue then showed that there exist Baire functions of every countable class and that there exist definable functions which are not analytically representable. He also defined the Borel measurable functions and showed that they too coincide with the Baire functions. In fact he proved a much stronger theorem along these lines which relates the hierarchy of Baire functions with a natural hierarchy of the Borel measurable sets at each level. Today we recognize Lebesgue [1905] as a classic work in the theory of definability. It introduced and studied systematically several natural notions of definable functions and sets and it established the first important hierarchy theorems and structure results for collections of definable objects. In it we can find the origins of many standard tools and techniques that we use today, for example universal sets and applications of the Cantor diagonal method to questions of definability.
1
2
Introduction
One of Lebesgue’s results in [1905] identified the implicitly analytically definable functions with the Baire functions. To take a simple case, suppose that f : R2 → R is analytically representable and for each x, the equation f(x, y) = 0 has exactly one solution in y. This equation then defines y implicitly as a function of x; Lebesgue showed that it is an analytically representable function of x, by an argument which was “simple, short but false.” The wrong step in the proof was hidden in a lemma taken as (basically) trivial, that a set in the line which is the projection of a Borel measurable set in the plane is itself Borel measurable. Ten years later the error was spotted by Suslin, then a young student of Lusin at the University of Moscow, who rushed to tell his professor in a scene charmingly described in Sierpinski [1950]. Suslin called the projections of Borel sets analytic and showed that indeed there are analytic sets which are not Borel measurable. Together with Lusin they quickly established most of the basic properties of analytic sets and they announced their results in two short notes in the Comptes Rendus, Suslin [1917] and Lusin [1917]. The class of analytic sets is rich and complicated but the sets in it are nice. They are measurable in the sense of Lebesgue, they have the property of Baire and they satisfy the Continuum Hypothesis, i.e., every uncountable analytic set is equinumerous with the set of all real numbers. The best result in Suslin [1917] is a characterization of the Borel measurable sets as precisely those analytic sets which have analytic complements. Lusin [1917] announced another basic theorem which implied that Lebesgue’s contention about implicitly analytically definable functions is true, despite the error in the original proof. Suslin died in 1919 and the study of analytic sets was continued mostly by Lusin and his students in Moscow and by Sierpinski in Warsaw. Because of what Lusin delicately called “difficulties of international communication” those years, they were isolated from each other and from the wider mathematical community, and there were very few publications in western journals in the early twenties. The next significant step was the introduction of projective sets by Lusin and Sierpinski in 1925: a set is projective if it can be constructed starting with Borel measurable sets and iterating the operations of projection and complementation. Using later terminology, let us call analytic sets A sets, analytic complements CA sets, projections of CA sets PCA sets, complements of these CPCA sets, etc. Lusin in his [1925a], [1925b], [1925c] and Sierpinski [1925] showed that these classes of sets are all distinct and they established their elementary properties. But it was clear from the very beginning that the theory of projective sets was not easy. There was no obvious way to extend to these more complicated sets the regularity properties of Borel and analytic sets; for example it was an open problem whether analytic complements satisfy the Continuum Hypothesis or whether PCA sets are Lebesgue measurable. Another fundamental and difficult problem was posed in Lusin [1930a]. Suppose P is a subset of the plane; a subset P ∗ of P uniformizes P if P ∗ is the graph of a function and it has the same projection on the line as P, as in the figure on the opposite page. The natural question is whether definable sets admit definable uniformizations and it comes up often, for example when we seek “canonical” solutions for y in terms of x in an equation f(x, y) = 0.
Introduction
3
P∗ P
Lusin and Sierpinski showed that Borel sets can be uniformized by analytic complements and Lusin also verified that analytic sets can be projectively uniformized. In a fundamental advance in the subject, Kondo [1938] completed earlier work of Novikov and proved that analytic complements and PCA sets can be uniformized by sets in the same classes. Again, there was no clear method for extending the known techniques to solve the uniformization problem for the higher projective classes. As it turned out, the “difficulties of the theory of projective sets” which bothered Lusin from his very first publication in the subject could not be overcome by ingenuity alone. There was an insurmountable technical obstruction to answering the central open questions in the field, since all of them were independent of the axioms of classical set theory. It goes without saying that the researchers in descriptive set theory were formulating and trying to prove their assertions within axiomatic Zermelo-Fraenkel set theory (with choice), as all mathematicians still do, consciously or not. ¨ The first independence results were proved by Godel, in fact they were by-products of his famous consistency proof of the Continuum Hypothesis. He announced in his [1938] that in the model L of constructible sets there is a PCA set which is not Lebesgue measurable: it follows that one cannot establish in Zermelo-Fraenkel set theory (with the Axiom of Choice and even if one assumes the Continuum Hypothesis) that all PCA sets are Lebesgue measurable. His results were followed up by some people, notably Mostowski and Kuratowski, but that was another period of “difficulties of international communication” and nothing was published until the late forties. Addison [1959b] gave the first exposition in print of the consistency and independence ¨ results that are obtained by analyzing Godel’s L. The independence of the Continuum Hypothesis was proved by Cohen [1963b], whose powerful method of forcing was soon after applied to independence questions in descriptive set theory. One of the most significant papers in forcing was Solovay [1970], where it is shown (among other things) that one can consistently assume the axioms of Zermelo-Fraenkel set theory (with choice and even the Continuum Hypothesis) together with the proposition that all projective sets are Lebesgue measurable; from ¨ this and Godel’s work it follows that in classical set theory we can neither prove nor disprove the Lebesgue measurability of PCA sets. Similar consistency and independence results were obtained about all the central problems left open in the classical period of descriptive set theory, say up to 1940. It says something about the power of the mathematicians working in the field those years, that in almost every instance they obtained the best theorems that could be proved from the axioms they were assuming. So the logicians entered the picture in their usual style, as spoilers. There was, however, another parallel development which brought them in more substantially and
4
Introduction
in a friendlier role. Before going into that, let us make a few remarks about the appropriate context for studying problems of definability of functions and sets. We have been recounting the development of descriptive set theory on the real numbers, but it is obvious that the basic notions are topological in nature and can be formulated in the context of more general topological spaces. All the important results can be extended easily to complete, separable, metric spaces. In fact, it was noticed early on that the theory assumes a particularly simple form on Baire space N = ù ù, the set of all infinite sequences of natural numbers, topologized with the product topology (taking ù discrete). The key fact about N is that it is homeomorphic with its own square N × N , so that irrelevant problems of dimension do not come up. Results in the theory are often proved just for N , with the (suitable) generalizations to other spaces and the reals in particular left for the reader or simply stated without proof. Let us now go back to a discussion of the impact of logic and logicians on descriptive set theory. ¨ The fundamental work of Godel [1931] on incompleteness phenomena in formal systems suggested that it should be profitable to delineate and study those functions (of several variables) on the set ù of natural numbers which are effectively computable. A great deal of work was done on this problem in the nineteen thirties by Church, ¨ Kleene, Turing, Post and Godel among others, from which emerged a coherent and beautiful theory of computability or recursion. The class of recursive functions (of several variables) on ù was characterized as the smallest set which contains all the constants, the successor and the projections (x1 , x2 , . . . , xn ) 7→ xi and which is closed under composition, a form of simple definition by induction (primitive recursion) and minimalization, where g is defined from f by the equation g(x1 , x2 , . . . , xn ) = least w such that f(x1 , x2 , . . . , xn , w) = 0, assuming that for each x1 , . . . , xn there is a root to the equation f(x1 , . . . , xn , w) = 0. Church [1936] and independently Turing [1936] proposed the Church-Turing Thesis (hypothesis) that all number theoretic functions which can be computed effectively by some algorithm are in fact recursive, and to this date no serious evidence has been presented to dispute this. Kleene [1952a], [1952b] extended the theory of recursion to functions f : ùn × N k → ù with domain some finite cartesian product of copies of the natural numbers and Baire space. For example, a function f : ù × N → ù is recursive (by the natural extension of the Church-Turing Thesis) if there is an algorithm which will compute f(n, α) given n and a sufficiently long initial segment of the infinite sequence α. A set A ⊆ ù n × N k is recursive if its characteristic function is recursive. By the Church-Turing Thesis again, these are the decidable sets for which we have (at least in principle) an algorithm for testing membership. Using recursion theory as his main tool, Kleene developed a rich and intricate theory of definability on the natural numbers in the sequence of papers [1943], [1955a], [1955b], [1955c].
Introduction
5
The class of arithmetical sets is the smallest family which contains all recursive sets and is closed under complementation and projection on ù. The analytical sets are defined similarly, starting with the arithmetical sets and iterating any finite number of times the operations of complementation, projection on ù and projection on N . Both these classes are naturally ramified into subclasses, much like the subclasses A, CA, PCA, . . . of projective sets of reals. Notice that the definitions make sense for subsets of an arbitrary product space of the form ù n × N k . Kleene, however, was interested in classifying definable sets of natural numbers and he stated his ultimate results just for them. The more complicated product spaces were brought in only so projection on N could be utilized to define complicated subsets of ù. Kleene studied a third notion (discovered independently by Davis [1950] and Mostowski [1951]) which is substantially more complicated. The class of hyperarithmetical sets of natural numbers is the smallest family of subsets of ù which contains the recursive sets and is closed under complementation and “recursive” countable union, suitably defined. The precise definition is quite intricate and the proofs of the main results are subtle, often depending on delicate estimates of the complexity of explicit and inductive definitions. Using later terminology, let us call Σ11 the simplest analytical sets of numbers, those which are projections to ù of arithmetical subsets of ù × N . The most significant result of Kleene [1955c] (and the whole theory for that matter) was a characterization of the hyperarithmetical sets as precisely those Σ11 sets which have Σ11 complements. Now this is clearly reminiscent of Suslin’s characterization of the Borel sets. A closer look at specific results reveals a deep resemblance between these two fundamental theorems and suggests the following analogy between the classical theory and Kleene’s definability theory for subsets of ù: R or N continuous functions Borel sets analytic sets projective sets
ù recursive functions hyperarithmetical sets Σ11 sets analytical sets.
In fact, the theories of the corresponding classes of objects in this table are so similar, that one naturally conjectures that Kleene was consciously trying to create an “effective analog” on the space ù of classical descriptive set theory. As it happened, Kleene did not know the classical theory, since he was a logician by trade and at the time that was considered part of topology. Mostowski knew it, being Polish, and he first used classical methods in his [1946], where he obtained independently many of the results of Kleene [1943]. More significantly, Mostowski introduced the hyperarithmetical sets following closely the classical approach to Borel sets, as opposed to Kleene’s initial rather different definition in his [1955b]. First to establish firmly the analogies in the table above was Addison, in his Ph.D. Thesis [1954] and later in his [1959a]. Over the years and with the work of many people, what was first conceived as “analogies” developed into a general theory which yields in a unified manner both the classical results and the theorems of the recursion theorists; more precisely, this effective theory yields refinements of the classical results and extensions of the theorems of the recursion theorists. It is this extended, effective descriptive set theory which concerns us here.
6
Introduction
Powerful as they are, the methods from logic and recursion theory cannot solve the “difficulties of the theory of projective sets,” since they too are restricted by the limitations of Zermelo-Fraenkel set theory. The natural next step was taken in the fundamental paper Solovay [1969], where for the first time strong set theoretic hypotheses were shown to imply significant results about projective sets. Solovay proved that if there exist measurable cardinals, then PCA sets are Lebesgue measurable, they have the property of Baire and they satisfy the Continuum Hypothesis. Later, he and Martin proved a difficult uniformization theorem about CPCA sets in their joint [1969], and Martin [1971] established several deep properties of CPCA sets, all under the same hypothesis, that there exist measurable cardinals. For our purposes here, it is not important to know exactly what measurable cardinals are. Suffice it to say that their existence cannot be shown in Zermelo-Fraenkel set theory and that if they exist, they are terribly large sets: bigger than the continuum, bigger than the first strongly inaccessible cardinal, bigger than the first Mahlo cardinal, etc. It is also fair to add that few people are willing to buy their existence after a casual look at their definition. Nevertheless, no one has shown that they do not exist, and it was known from previous work of Scott, Gaifman, Rowbottom and Silver that the existence of measurable cardinals implies new and interesting propositions about sets, even about real numbers. These, however, were metamathematical results, the kind that only logicians can love. Solovay’s chief contribution was that he used this new and strange hypothesis to solve natural, mathematical problems posed by Lusin more than forty years earlier. Unfortunately, measurable cardinals were not a panacea. Soon after Solovay’s original work it was shown by himself, Martin and Silver among others that they do not resolve the open questions about projective sets beyond the CPCA class, except for some isolated results about PCPCA sets. The next step was quite unexpected, even by those actively searching for strong hypotheses to settle the old open problems. Blackwell [1967] published a new, short and elegant proof of an old result of Lusin’s about analytic sets, using the determinacy of open games. Briefly, an infinite game (of perfect information) on ù is described by an arbitrary subset A ⊆ N of Baire space. We imagine two players I and II successively choosing natural numbers, with I choosing k0 , then II choosing k1 , then I choosing k2 , etc.; after an infinite sequence α = (k0 , k1 , . . . ) has been specified in this manner, we say that I wins if α ∈ A, II wins if α ∈ / A. The game (or the set A which describes it) is determined, if one of the two players has a winning strategy, a method of playing against arbitrary moves of his opponent which will always produce a sequence winning for him. It was known that open games are determined and Blackwell’s proof hinged on that fact. It was also known that one could prove the existence of non-determined games using the Axiom of Choice, but no definable non-determined game on ù had ever been produced. Working independently, Addison and Martin realized that Blackwell’s proof could be lifted to yield new results about the third class of projective sets, if only one assumed the hypothesis that enough projective sets are determined. Soon after, Martin and Moschovakis again independently used the hypothesis of projective determinacy to settle a whole slew of old questions about all levels of the projective hierarchy,
Introduction
7
see Addison and Moschovakis [1968] and Martin [1968]. Three years later the uniformization problem was solved on the same hypothesis in Moschovakis [1971a] and the methods introduced there led quickly to an almost complete structure theory for the classes of projective sets, see especially Kechris [1973], [1974], [1975], Martin [1971] and Moschovakis [1973], [1974c]. This is where matters stand today.
CHAPTER 1
THE BASIC CLASSICAL NOTIONS
Let ù = {0, 1, 2, . . . } be the set of (nonnegative) integers and let R be the set of real numbers. The main business of Descriptive Set Theory is the study of ù, R and their subsets, with particular emphasis on the definable sets of integers and reals. Another fair name for it is Definability Theory for the Continuum. In this first chapter we will introduce some of the basic notions of the subject and we will establish the elementary facts about them.
1A. Perfect Polish spaces Instead of working specifically with the reals, we will frame our results in the wider context of complete, separable metric spaces (Polish spaces) with no isolated points (perfect). One of the reasons for doing this is the wider applicability of the theory thus developed. More than that, we often need to look at more complicated spaces in order to prove results about R.(1–5) Of course R is a perfect Polish space and so is the real n-space Rn for each n ≥ 2. There are two other important examples of such spaces which will play a key role in the sequel. Baire space is the set of all infinite sequences of integers (natural numbers), N = ùù with the natural product topology, taking ù discrete. The basic neighborhoods are of the form N (k0 , . . . , kn ) = {α ∈ N : α(0) = k0 , . . . , α(n) = kn }, one for each tuple k0 , . . . , kn . We picture N as (the set of infinite branches of) a tree, where each node splits into countably many one-point extensions, Figure 1A.1. It is easy to verify that the topology of N is generated by the metric 0, if α = â, d (α, â) = 1 , if α 6= â. least n[α(n) 6= â(n)] + 1
Also, N is complete with this metric and the set of ultimately constant sequences is countable and dense in N , so N is a perfect Polish space. One can show that N is homeomorphic with the set of irrational numbers, topologized as a subspace of R. The proof appeals to some basic properties of continued fractions and does not concern us here—it can be found in any good book on number theory, for example Hardy and Wright [1960]. Although we will never use this result, we will find it convenient to call the members of N irrationals. 9
10
1. The basic classical notions
0
1
0
0
1 2
1
0
0
2
1
2
3
0
1
[1A.1
3
2
2
0
1
···
0
1
3
Figure 1A.1. Picturing N as a tree. Notice that Baire space is totally disconnected, i.e., the neighborhood base given above consists of clopen (closed and open) sets. 1A.1. Theorem. For every Polish space M, there is a continuous surjection ð:N ։M of Baire space onto M. Proof. Fix a countable dense subset D = {r0 , r1 , r2 , . . . } of M and to each α ∈ N assign the sequence {xnα } = {xn } by the recursion x0 = rα(0) ( r xn+1 = α(n+1) xn
if d (xn , rα(n+1) ) < 2−n , if d (xn , rα(n+1) ) ≥ 2−n .
Now for each n, d (xn , xn+1 ) < 2−n , so {xnα } is Cauchy and we can set ð(α) = limn→∞ xnα . It is obvious that ð is continuous since α(0) = â(0), . . . , α(n) = â(n) =⇒ x0α = x0â , . . . , xnα = xnâ from which it follows immediately that d ð(α), ð(â) ≤ d ð(α), xnα + d xnâ , ð(â) ≤ 2−n+1 + 2−n+1 = 2−n+2 .
On the other hand, for each x ∈ M let α(n) = least k such that d (x, rk ) < 2−n−1
1A.2]
1A. Perfect Polish spaces
11
0
0 0
0
1 1
0
1 1/3
2/3
1
1
Figure 1A.2. The Cantor set. and check that ð(α) = limn rα(n) = x. ⊣ Another very useful perfect Polish space is the set of all infinite binary sequences C = ù 2, again with the product topology. This is a compact subspace of N naturally represented by the complete binary tree. It is obviously homeomorphic with the classical Cantor set obtained from the closed interval [0, 1] on the line by successively removing the open middle third, as in Figure 1A.2. Again we will abuse terminology a bit by calling C the Cantor set. With each perfect Polish space M we can associate a fixed enumeration N (M, 0), N (M, 1), N (M, 2), . . . of a countable set of open nbhds which generates the topology. When M is clearly understood by the context we will use the simpler notation N0 , N1 , N2 , . . . . Of course we may assume that the Ni ’s are open balls. There are situations, however, when this is not convenient. For example, if M = X1 × X2 is the product of two spaces, it is often preferable to work with the nbhds of the form B1 × B2 , where B1 and B2 are chosen from bases in X1 and X2 . We will leave open the possibility that the Ni ’s are not open balls. However, we will assume that with each Ni we have associated a center xi and a radius pi such that the following hold: (1) xi ∈ Ni , if Ni 6= ∅. (2) If x ∈ Ni , then d (x, xi ) < pi . (3) If x is any point, then for every n we can find some Ni such that x ∈ Ni and radius(Ni ) < 2−n . For any set P ⊆ M, let P = closure of P, so that N s = N (M, s) is the closure of the s’th nbhd in the fixed base for the topology for M. The simple construction in the next result will be useful in many situations beyond the corollary following it. 1A.2. Theorem. Let M be a perfect, Polish space. We can assign to each finite binary sequence u = (t0 , . . . , tn−1 ) (ti = 0, 1) an open nbhd Nó(u) 6= ∅ in M so that
12
1. The basic classical notions
[1A.3
Nó(∅) Nó(0) Nó(0,0)
Nó(1) Nó(0,1)
Figure 1A.3. (i) if u is a proper initial segment of v, then N ó(v) ⊆ Nó(u) , (ii) if u an v are incompatible, then N ó(u) ∩ N ó(v) = ∅, (iii) if u = (t0 , . . . , tn−1 ) has length n, then radius(Nó(u) ) ≤ 2−n . (See Figure 1A.3.) Proof. Two sequences u = (t0 , . . . , tn−1 ), v = (s0 , . . . , sk−1 ), are incompatible, if for some i < n, i < k we have ti 6= si . We define Nó(u) by induction on the length n of the binary sequence u = (t0 , . . . , tn−1 ) starting with some Nó(∅) of radius ≤ 1 = 2−0 that we assign to the empty sequence. Given u = (t0 , . . . , tn−1 ) and assuming that Nó(u) has already been defined, we know that there must be infinitely many points in Nó(u) or else the center of this nbhd would be isolated. Choose then x 6= y in Nó(u) and find open balls Bx , By with centers x and y respectively and such that B x ⊆ Nó(u) ,
B y ⊆ Nó(u) ,
B x ∩ B y = ∅, as in Figure 1A.4. It is now enough to choose i, j such that Ni ⊆ Bx , Nj ⊆ By and Ni , Nj have radii ≤ 2−n−1 and set ó(t0 , . . . , tn−1 , 0) = i,
ó(t0 , . . . , tn−1 , 1) = j.
Verification of (i), (ii) and (iii) with this definition of ó is trivial. ⊣ 1A.3. Corollary. For every perfect Polish space M, there is a continuous injection ð:CM of the Cantor set into M. Proof. Given an infinite binary sequence α, put xnα = the center of N M, ó α(0), . . . , α(n − 1)
and let
ð(α) = limn→∞ xnα .
1B]
1B. The Borel pointclasses of finite order
x
13
Nó(u)
Ni y Nj
Figure 1A.4. It is immediate that ð is an injection (one-to-one). That ð is continuous can be proved by verifying o S n ð−1 [Ns ] = n α : N (M, ó(α(0), . . . , α(n − 1))) ⊆ Ns ⊣
Exercises 1A.4. For each compact Polish space X , let C [X ] be the set of all continuous functions on X to R with the usual supnorm distance, d (f, g) = supremum{|f(x) − g(x)| : x ∈ X }. Prove that C [X ] is a perfect Polish space. Hint. Use the separability of X and appeal to the Stone-Weierstrass Theorem. ⊣ 1A.5. For each perfect Polish space X , let H [X ] be the set of all compact non-empty subsets of X . If x ∈ X and A ∈ H [X ], put d (x, A) = infimum{d (x, y) : y ∈ A} where on the right d is the distance function on X . The Hausdorff distance between two compact sets is defined by d (A, B) = maximum supremum{d (x, B) : x ∈ A}, supremum{d (y, A) : y ∈ B} . Prove that this is a metric on H [X ] and that H [X ] is a perfect Polish space. Hint. The set of all finite subsets of any dense subset of X is dense in H [X ].
⊣
1B. The Borel pointclasses of finite order In order to study the subsets of a perfect Polish space M, it will be necessary to consider other spaces related to M, e.g., the products M × M, N × M, ù × M. Let us first establish notation and terminology which make these detours easy. We fix once and for all a collection F of metric spaces with the following properties:
14
1. The basic classical notions
[1B
Y
(x, y)
N = {(x ′ , y ′ ) : d (x, y), (x ′ , y ′ ) < 1}
X Figure 1B.1. The unit ball in a product space. (1) The discrete space ù, the reals R, Baire space N and the Cantor set C are in F. (2) Every space in F other than ù is a perfect Polish space. Except for these restrictions we can leave membership in F open—e.g., one might take ù, R, N and C to be the only spaces in F. The idea is that we put in F all the perfect Polish spaces in which we are interested. The members of F are the basic spaces. A product space (by definition) is any cartesian product X = X1 × · · · × Xk , where each Xi is basic. Basic spaces count as product spaces by allowing k = 1 here. We naturally topologize X1 × · · · × Xk as a product, i.e., with basic nbhds of the form N = B1 × · · · × Bk , where each Bi is a nbhd in Xi . It is easy to verify that this topology on X is induced by the metric d (x1 , . . . , xk ), (y1 , . . . , yk ) = maximum{d1 (x1 , y1 ), . . . , dk (xk , yk )}, where each di is the given metric on Xi . (See Figure 1B.1.) Two product spaces X = X1 × · · · × Xk and Y = Y1 × · · · × Yl are equal if k = l and X1 = Y1 , . . . , Xk = Yk . We then define products of product spaces by going back to the basic factors, i.e., if X = X1 × · · · × Xk and Y = Y1 × · · · × Yl , then (by definition) X × Y = X1 × · · · × Xk × Y1 × · · · × Yl . Thus X × (Y × Z) = (X × Y) × Z = X × Y × Z. We call the tuples in these product spaces points and the subsets of these spaces pointsets. If x = (x1 , . . . , xn ) and y = (y1 , . . . , yl ) then (by definition) (x, y) = (x1 , . . . , xk , y1 , . . . , yl ). As with products of product spaces, this pairing operation is associative, x, (y, z) = (x, y), z = (x, y, z).
1B]
1B. The Borel pointclasses of finite order
15
ù
P
∃ù P
X
Figure 1B.2. Projection along ù. We think of pointsets as sets or as relations with arguments in the basic spaces. Both points of view are useful and we will use interchangeably the customary notations for these, i.e., for P ⊆ X , x ∈ P ⇐⇒ P(x). Of course we will not be studying individual pointsets so much as collections of pointsets, call them pointclasses. Thus a pointclass Λ is a collection of sets such that each P in Λ is a subset of some product space X . For example, we may have Λ = all open pointsets = {P : P ⊆ X for some product space X and P is open.} In definability theory we typically start with a small pointclass Λ and certain operations on pointsets and then we study the sets which can be constructed by applying (once or repeatedly) the given operations to the members of Λ. For the Borel sets of finite order we start with the open sets and we apply repeatedly the operations of complementation or negation (¬) and projection along ù or existential number quantification (∃ù ). More precisely, if P ⊆ X is any pointset, put ¬P = X \ P. For a pointclass Λ, let ¬Λ = {¬P : P ∈ Λ} be the dual pointclass. Similarly, if P ⊆ X × ù for some X , let ∃ù P = {x ∈ X : for some n, P(x, n)} = {x ∈ X : (∃n)P(x, n)} and for a pointclass Λ put ∃ù Λ = {∃ù P : P ∈ Λ, P ⊆ X × ù for some X }; see Figure 1B.2. The Borel pointclasses of finite order Σ 0n (n ≥ 1) are defined by the recursion e Σ 01 = all open pointsets, e Σ 0n+1 = ∃ù ¬Σ 0n ; e e
16
1. The basic classical notions
⊆ ∆ 01 e
⊆
Σ 01 e
⊆
⊆
⊆
Π 01 e
∆ 02 e
⊆
Σ 02 e Π 02 e
[1B.1
⊆ ⊆
⊆ ∆ 03 e
Σ 03 e
···
···
⊆
Π 03 · · · e
Diagram 1B.3. The Borel pointclasses of finite order.
the dual Borel pointclasses Π 0n are defined by e Π 0n = ¬Σ 0n ; e e finally, the ambiguous Borel pointclasses ∆ 0n are given by(13) e ∆ 0n = Σ 0n ∩ Π 0n . e e e Thus, Π 01 consists of all closed pointsets, ∆ 01 is the class of all clopen sets, Σ 02 is the e projections along ù of closed sets, e etc. Put another way, a set Peis Σ 0 if class of all 2 e there is a closed F ⊆ X × ù such that for all x, P(x) ⇐⇒ (∃t)F (x, t).
Similarly, P is
Σ 03 e
if there is a closed F such that
P(x) ⇐⇒ (∃t)¬(∃s)F (x, t, s) ⇐⇒ (∃t)(∀s)¬F (x, t, s),
i.e., P is Σ 03 if there is an open pointset G such that e P(x) ⇐⇒ (∃t1 )(∀t2 )G(x, t1 , t2 ).
Similar normal forms can be computed for the pointclasses Π 0n , e.g., P is Π 04 if there e e is some open G such that for all x, P(x) ⇐⇒ (∀t1 )(∃t2 )(∀t3 )G(x, t1 , t2 , t3 ).
In the classical terminology, Σ 02 sets are called Fó sets, Π 02 sets are Gä , Σ 03 sets are Gäó , Π 03 sets are Fóä , etc. It is aecumbersome notation ande we will not usee it, except for aneoccasional reference to Fó ’s and Gä ’s. 1B.1. Theorem. The diagram of inclusions 1B.3 holds among the Borel pointclasses of finite order. Proof. The inclusions Σ 0n ⊆ Π 0n+1 e e are almost immediate from the definitions. Taking n = 3 to simplify notation, if P is Σ 03 , then e P(x) ⇐⇒ (∃t1 )(∀t2 )G(x, t1 , t2 ) with some open G ⊆ X × ù × ù. We can rewrite this as (∗)
P(x) ⇐⇒ (∀s)(∃t1 )(∀t2 )G(x, t1 , t2 )
since the addition of the vacuous quantifier (∀s) does not affect the meaning of the equivalence. Now define G ′ (x, s, t1 , t2 ) ⇐⇒ G(x, t1 , t2 ) and notice that G ′ is (trivially) open, so equivalence (∗) above establishes that P is Π 04 . e
1B.5]
1B. The Borel pointclasses of finite order
17
To prove the inclusions Σ 0n ⊆ Σ 0n+1 , e e recall that in a separable metric space every open set is a countable union of closed sets. If G ⊆ X and S G = t Ft
with each Ft closed, define F ⊆ X × ù by
F (x, t) ⇐⇒ x ∈ Ft and notice that F is closed and G(x) ⇐⇒ (∃t)F (x, t). Thus G is Σ 02 , and since it was arbitrary open, e Σ 01 ⊆ Σ 02 . e e Hence Σ 02 = ∃ù ¬Σ 01 ⊆ ∃ù ¬Σ 02 = Σ 03 and inductively, Σ 0n ⊆ Σ 0n+1 . This establishes e e e e e e Σ 0n ⊆ ∆ 0n+1 e e for every n, so taking negations, Π 0n ⊆ ∆ 0n+1 e e and the remaining inclusions in the diagram are trivial.
⊣
Exercises 1B.2. Prove that if X = X1 × · · · × Xk is a product space with at least one factor Xi = N and every Xj either ù or N , the X is homeomorphic with N . Hint. Construct homeomorphisms of ù × N and N × N with N and then use induction on k. ⊣ 1B.3. Prove that if X = X1 × · · · × Xk is a product space with at least one factor Xi not ù, then X is a perfect Polish space. 1B.4. Prove that a pointset P is Σ 02 if and only if e S P= ∞ i=0 Fi ,
with each Fi closed. Similarly, P is Π 02 , if and only if e with each Gi open.
P=
T∞
i=0
Gi
This is the classical definition of Fó and Gä sets. These occur quite often in analysis, for example consider the following problem. 1B.5. Let f : R → R be an arbitrary function on the line. Prove that the set A = {x ∈ R : f is continuous at x} is a Gä .
18
1. The basic classical notions
[1B.6
Hint. Define the variation of f on an interval (a, b) by V (a, b) = supremum{f(x) : a < x < b} − infimum{f(x) : a < x < b}, where the value may be ∞ or −∞. The local variation of f is given by v(x) = limn→∞ V x − n1 , x + n1
and it is clear that f is continuous at x just in case v(x) = 0. Show that for each n, the set n 1o An = x : v(x) < n T is open and A = n An . ⊣ 1B.6. Prove that if n ≥ 3 is odd, then P is Σ 0n if and only if there is an open set G e such that P(x) ⇐⇒ (∃t1 )(∀t2 )(∃t3 )(∀t4 ) · · · (∀tn−1 )G(x, t1 , . . . , tn−1 ).
Similarly, if n is even then P is Σ 0n is and only if there is a closed set F such that e P(x) ⇐⇒ (∃t1 )(∀t2 )(∃t3 ) · · · (∃tn−1 )F (x, t1 , . . . , tn−1 ).
Find similar normal forms for the Π 0n pointsets. e 1B.7. Prove that if X is a product of copies of ù and N and P is Σ 0n with n odd, e then there exists a clopen set R such that P(x) ⇐⇒ (∃t1 )(∀t2 ) · · · (∀tn−1 )(∃tn )R(x, t1 , . . . , tn );
similarly for even n, with the last quantifier ∀. S Hint. If A ⊆ X is open, then A = n Rn with clopen Rn in these spaces and we can take R(x, n) ⇐⇒ x ∈ Rn .
⊣
1C. Computing with relations; closure properties The relational notation for pointsets is particularly convenient for putting down compact expressions for complicated definitions. Suppose, for example, that Q ⊆ X × N , R ⊆ X × N × ù and let P(x) ⇐⇒ (∀α)[Q(x, α) =⇒ (∃i)R(x, α, i)]. Here the logical symbols are taken with their customary meaning, as we have been using them all along: ∀ (for all), ∃ (there exists), =⇒ (implies), & (and), ∨ (or), ¬ (not). We will also use customarily Greek variables α, â, ã, . . . from the beginning of the alphabet over N and i, j, k, l, m, n, s, t over ù. This will save us having to specify explicitly the range of the quantifiers in each definition. One can view the logical symbols as denoting operations on pointsets. In general, a k-ary pointset operation is a function Φ with domain some set of k-tuples of pointsets and pointsets as values. With this terminology, conjunction & is the binary pointset operation which assigns to every pair P, Q of subsets of the same space X the set P & Q, x ∈ (P & Q) ⇐⇒ P(x) & Q(x).
1C]
1C. Computing with relations; closure properties
19
Y
P
∃Y P
X
Figure 1C.1. Projection along Y. Of course P & Q = P ∩ Q. We will however keep the symbol ∩ for denoting the general set theoretic operation of intersection, with A ∩ B defined for arbitrary sets A, B. Similarly, the disjunction P ∨ Q of two pointsets is defined when P and Q are subsets of the same X and P ∨ Q = P ∪ Q = {x : P(x) ∨ Q(x)}. Negation is most conveniently regarded as a collection of operations ¬X , one for each product space X , with ¬X P defined when P ⊆ X : ¬X P = X \ P = {x ∈ X : ¬P(x)}. In practice we will always write ¬P for ¬X P, as X is clear from the context. From these we can construct more pointset operations by composition, e.g., the implication P =⇒ Q of P and Q is defined by (P =⇒ Q) = ¬P ∨ Q. More interesting than these propositional pointset operations are the projections and dual projections or quantifiers. If P ⊆ X × Y, put ∃Y P = {x ∈ X : (∃y)P(x, y)} as in Figure 1C.1. For each fixed product space Y, we call the operation ∃Y projection along Y or existential quantification on Y. Clearly ∃Y P is defined when P ⊆ X × Y for some X , in which case ∃Y P ⊆ X . We have already used projection along ù, ∃ù . Only the projections along basic spaces are fundamental, since all the others can be obtained from these by composition; for example, if Y = ù × N , then for each P ⊆ X × ù × N, ∃Y P = ∃ù ∃N P, i.e., in relational notation, (∃y ∈ Y)P(x, y) ⇐⇒ (∃n)(∃α)P(x, n, α). If P ⊆ X × Y, put ∀Y P = ¬∃Y ¬P,
20
1. The basic classical notions
[1C.1
Y P
¬P
¬P
∀Y P
∀Y P
X
Figure 1C.2. Universal quantification on Y. i.e., x ∈ ∀Y P ⇐⇒ ¬(∃y ∈ Y)¬P(x, y) ⇐⇒ (∀y ∈ Y)P(x, y). Fixing Y, we call the operation ∀Y dual projection along Y or universal quantification on Y. Again ∀Y P is defined when P ⊆ X × Y for some X and then ∀Y P ⊆ X , see Figure 1C.2. In addition to the operations ∃ù , ∀ù , the bounded number quantifiers will prove useful, (x, n) ∈ ∃≤ P ⇐⇒ (∃m ≤ n)P(x, m), (x, n) ∈ ∀≤ P ⇐⇒ (∀m ≤ n)P(x, m), see Figure 1C.3. Clearly ∃≤ P, ∀≤ P are defined when P ⊆ X × ù for some X , in which case both ≤ ∃ P and ∀≤ P are also subsets of X × ù. A pointclass Λ is closed under a k-ary pointset operation Φ if whenever P1 , . . . , Pk are in Λ and Φ(P1 , . . . , Pk ) is defined, then Φ(P1 , . . . , Pk ) is also in Λ. For example, Λ is closed under conjunction if whenever P and Q are subsets of the same space X and both are in Λ, then P & Q = P ∩ Q is in Λ. Similarly, Λ is closed under negation ¬, if ¬Λ ⊆ Λ, i.e., for every P ∈ Λ, P ⊆ X we have X \ P ∈ Λ. We say that Λ is closed under continuous substitution if for every continuous function f : X → Y and every P ∈ Λ, P ⊆ Y, f −1 [P] ∈ Λ. Here of course x ∈ f −1 [P] ⇐⇒ f(x) ∈ P ⇐⇒ P f(x) . It is worth putting down a very useful alternative version of this closure property. 1C.1. Lemma. Suppose Λ is a pointclass closed under continuous substitution, let f1 : X → Y1 , . . . , fm : X → Ym be continuous functions and assume that Q ⊆ Y1 ×· · ·×Ym is a pointset in Λ. If P(x) ⇐⇒ Q f1 (x), . . . , fm (x) , then P is also in Λ. Proof. The function g : X → Y1 × · · · × Ym defined by g(x) = f1 (x), . . . , fm (x)
1C.1]
1C. Computing with relations; closure properties ù
21
∃≤ P P
X Figure 1C.3. Bounded number quantification. is continuous and
For example, suppose
P(x) ⇐⇒ Q g(x) .
⊣
P(x, y) ⇐⇒ Q(y, x) & R(x, y, y), where Q ⊆ Y × X , R ⊆ X × Y × Y and both Q and R are in some pointclass Λ closed under continuous substitution and & . Then P too is in Λ, since P(x, y) ⇐⇒ Q ′ (x, y) & R′ (x, y), where Q ′ (x, y) ⇐⇒ Q f1 (x, y), f2 (x, y) , with
R′ (x, y) ⇐⇒ R f2 (x, y), f1 (x, y), f1 (x, y) f1 (x, y) = y,
f2 (x, y) = x.
In effect, closure under continuous substitution allows us to permute or identify variables in a relation and stay in the pointclass we are working with. After these preliminary remarks we can state concisely the elementary closure properties of the Borel classes. To prove them, we will need functions that code finite sequences of integers by single integers. Let p(i) = pi = the i’th prime, with p0 = 2, and for each n, put t
+1
n−1 ht0 , . . . , tn−1 i = p0t0 +1 · · · · · pn−1 .
By convention the empty product is 1, so that h∅i = 1,
22
1. The basic classical notions
[1C.2
and 1 is the code of the empty sequence. With this particular coding of tuples we associate the natural decoding functions and relations Seq(u) ⇐⇒ u = 1 or u = ht0 , . . . , tn−1 i for some t0 , . . . , tn−1 , ( n if u = ht0 , . . . , tn−1 i for some t0 , . . . , tn−1 , lh(u) = 0 otherwise, ( ti if u = ht0 , . . . , tn−1 i for some t0 , . . . , tn−1 and i < n, (u)i = 0 otherwise. It is often convenient to index a finite sequence starting with 1 rather than 0. Notice that if u = ht1 , . . . , tn i, then for i < n, (u)i = ti+1 . 1C.2. Theorem. Each Borel pointclass Σ 0n (n ≥ 1) is closed under continuous substie tution, ∨, & , ∃≤ , ∀≤ and ∃ù . 0 Each dual pointclass Π n is closed under continuous substitution, ∨, & , ∃≤ , ∀≤ and ∀ù . e pointclass ∆ 0 is closed under continuous substitution, ¬, ∨, & , Each ambiguous Borel n ≤ ≤ e ∃ and ∀ . Proof. The results about Π 0n and ∆ 0n follow immediately from those about Σ 0n . The e trivial, e except perhaps for closure under ∃≤ eand ∀≤ closure properties of Σ 01 are also e the equations which follow easily from S ∃≤ P = n {(x, n) : (∃m ≤ n)P(x, m)}, S ∀≤ P = n {(x, n) : (∀m ≤ n)P(x, m)}. Assume now that Σ 0n has all the right closure properties—we will show the same for e
Σ 0n+1 .
e Suppose first that Q is a typical Σ 0 subset of Y, i.e., n+1 e Q(y) ⇐⇒ (∃m)¬P(y, m),
with P some Σ 0n subset of Y × ù. Assume also that f : X → Y is continuous. Now e Q f(x) ⇐⇒ (∃m)¬P f(x), m ⇐⇒ (∃m)¬P ′ (x, m)
with
P ′ (x, m) ⇐⇒ P f(x), m .
Since P ′ is Σ 0n by 1C.1 and the induction hypothesis, f −1 [Q] is Σ 0n+1 . Hence Σ 0n+1 is e continuous substitution. e e closed under 0 To prove closure of Σ n+1 under & , compute e R(x) ⇐⇒ (∃s)¬P(x, s) & (∃t)¬Q(x, t) ⇐⇒ (∃u) ¬P x, (u)0 & ¬Q x, (u)1 ⇐⇒ (∃u)¬ P x, (u)0 ∨ Q x, (u)1 .
If P and Q are in Σ 0n , then e P ′ (x, u) ⇐⇒ P x, (u)0 ∨ Q x, (u)1
is also Σ 0n by closure under continuous substitution and ∨, so R is Σ 0n+1 . e e
1C.2]
1C. Computing with relations; closure properties
23
This method of proof goes by the fancy name of like quantifier contraction and yields equally trivial proofs of closure of Σ 0n+1 , under ∨ and ∃ù . For closure under ∀≤ we e need a slightly more elaborate contraction of finitely many quantifiers. Suppose R(x, n) ⇐⇒ (∀m ≤ n)(∃s)¬P(x, m, s)
with P in Σ 0n and compute, e R(x, n) ⇐⇒ (∃u)(∀m ≤ n)¬P x, m, (u)m ⇐⇒ (∃u)¬(∃m ≤ n)P x, m, (u)m .
Again
P ′ (x, n, u) ⇐⇒ (∃m ≤ n)P x, m, (u)m
is Σ 0n by closure of this class under continuous substitution and ∃≤ , so R is Σ 0n+1 . e e Proof of closure of Σ 0n+1 under ∃≤ is trivial. ⊣ e This simple argument is a good illustration of the advantage of relational (or logical) notation, i.e., writing R(x, n) ⇐⇒ (∀m ≤ n)(∃s)¬P(x, m, s) rather than R = ∀≤ ∃ù ¬P. In fact the whole proof rested on some quantifier manipulation rules whose truth is transparent in logical notation. We list them here for reference, but we will apply them in the future without much ado. (∃s)(∃t)P(s, t) ⇐⇒ (∃u)P (u)0 , (u)1 , (∀s)(∀t)P(s, t) ⇐⇒ (∀u)P (u)0 , (u)1 (∀m ≤ n)(∃s)P(m, s) ⇐⇒ (∃u)(∀m ≤ n)P m, (u)m , (∃m ≤ n)(∀s)P(m, s) ⇐⇒ (∀u)(∃m ≤ n)P m, (u)m .
These rules are useful because they allow us to simplify the quantifier prefix of a complicated logical expression by introducing continuous substitutions in the matrix. To see how one can use the closure properties of a pointclass, suppose that P(x) ⇐⇒ (∃t)(∃s) Q(x, s) =⇒ (∃u) R u, f(x, u), t ∨ S(u, x, s) ,
where Q, R, S are Σ 0n , f is continuous and t, s, u range over ù. We will argue that P e is in Σ 0n+1 . e First put Q ′ (x, t, s) ⇐⇒ Q(x, s),
R′ (x, t, s, u) ⇐⇒ R u, f(x, u), t , S ′ (x, t, s, u) ⇐⇒ S(u, x, s),
24
1. The basic classical notions [1C.3 P(x) ⇐⇒ (∃s)(∃t) Q(x, s) =⇒ (∃u) R u, f(x, u), t ∨ S(u, x, s) ⇐⇒ (∃s)(∃t) ¬Q(x, s) ∨ (∃u) R u, f(x, u), t ∨ S(u, x, s) Π 0n e
Σ 0n+1 e Σ 0n+1 e
Σ 0n e
Σ 0n+1 e
Σ 0n e Σ 0n+1 e
Σ 0n e
Σ 0n e
Diagram 1C.4. and notice that Q ′ , R′ , S ′ are Σ 0n by closure of this pointclass under continuous e substitution. Now P(x) ⇐⇒ (∃t)(∃s){¬Q ′ (x, t, s) ∨ (∃u)[R′ (x, t, s, u) ∨ S ′ (x, t, s, u)]} ⇐⇒ (∃t)(∃s){¬Q ′ (x, t, s) ∨ (∃u)T (x, t, s, u)} ⇐⇒ (∃t)(∃s){¬Q ′ (x, t, s) ∨ T ′ (x, t, s)} ⇐⇒ (∃t)(∃s)T ′′ (x, t, s) where T , T ′ , T ′′ are defined by T (x, t, s, u) ⇐⇒ R′ (x, t, s, u) ∨ S ′ (x, t, s, u), T ′ (x, t, s) ⇐⇒ (∃u)T (x, t, s, u), T ′′ (x, t, s) ⇐⇒ ¬Q ′ (x, t, s) ∨ T ′ (x, t, s). Clearly T and T ′ are Σ 0n by the closure properties of this pointclass. Hence T ′ is Σ 0n+1 e by 1B.1 and since ¬Q eis also Σ 0n+1 , T ′′ is Σ 0n+1 . Finally P is Σ 0n+1 by two applications ù e e of closure of this pointclass under ∃ . e
This kind of computation is so simple that we will not usually bother to put it down. One way to make computations of this type with a minimum of writing is to use a diagram like 1C.4 which shows step-by-step the properties of the relevant pointclasses that we use.
Exercises 1C.3. Let f : R → R be a continuous function on the line. Prove that the relations P(x, y) ⇐⇒ f ′ (x) = y, Q(x) ⇐⇒ f ′ (x) exists are both Π 03 . e Hint. Let r0 , r1 , . . . be an enumeration of all rational numbers and put 1 1 R(x, y, s, k, m) ⇐⇒ rm 6= 0 & {f(x + rm ) − f(x)} − y ≤ . rm s +1
1C.6]
1C. Computing with relations; closure properties Y
Px
25
P∗ P
x
X
Figure 1C.5. Uniformization. Clearly R is a closed relation. It is easy to verify that n o 1 P(x, y) ⇐⇒ (∀s)(∃k)(∀m) 0 < |rm | < =⇒ R(x, y, s, k, m) . k+1 The second assertion is proved similarly, starting with the relation S(x, s, k, m, n) ⇐⇒ rm 6= 0 & rn 6= 0 & r1m {f(x + rm ) − f(x)} − r1n {f(x + rn ) − f(x)} ≤
1 s+1 .
⊣
1C.4. Let C [0, 1] be the space of continuous real functions on the unit interval and define Q ⊆ C [0, 1] × R by Q(f, x) ⇐⇒ 0 < x < 1 & f ′ (x) exists. Prove that Q is Π 03 . e 1C.5. Prove that if P ⊆ X and Q ⊆ Y are Σ 0n then the product P × Q ⊆ X × Y is e Σ 0n . e Hint. Use closure under continuous substitution. ⊣
For the next exercise we introduce the basic problem of uniformization.(17) Suppose P ⊆ X × Y is a subset of the product X × Y. We say that P ∗ uniformizes P, if P ∗ ⊆ P and P ∗ is the graph of a function with domain the projection ∃Y P. Intuitively, P ∗ assigns to each point in ∃Y P just one member of the section or fiber Px = {y : P(x, y)} as in Figure 1C.5. It follows from the axiom of choice that each P can be uniformized by some P ∗ ; on the other hand, it is often very difficult to find a definable uniformizing set, even if the given set is very simple. The next exercise solves the uniformization problem in a very simple situation, but we will see later that even this easy result is useful. 1C.6. Prove that for each n > 1, if P ⊆ X × ù is in Σ 0n , then there is some P ∗ also e in Σ 0n which uniformizes P. e Hint. Suppose P(x, m) ⇐⇒ (∃i)Q(x, m, i)
26
1. The basic classical notions
[1C.7
Q P
-
Q∗ P∗ Figure 1C.6. Reduction. with Q in Π 0n−1 . Put e R(x, s) ⇐⇒ Q x, (s)0 , (s)1 & (∀t < s)¬Q x, (t)0 , (t)1 , P ∗ (x, m) ⇐⇒ (∃i)R(x, hm, ii).
⊣
Suppose P and Q are subsets of the same space X . We say that the pair P ∗ , Q ∗ reduces the pair P, Q if the following hold:(16) P ∗ ⊆ P,
Q∗ ⊆ Q
P ∪ Q = P ∗ ∪ Q∗, P ∗ ∩ Q ∗ = ∅. (See Figure 1C.6.) 1C.7. Prove that for each n > 1, every pair of sets P, Q in Σ 0n is reducible by a pair e P , Q ∗ in Σ 0n . e Hint. Uniformize the set R defined by ∗
R(x, m) ⇐⇒ {P(x) & m = 0} ∨ {Q(x) & m = 1}.
⊣
Suppose that P and Q are disjoint subsets of the same space X . We say that the set S separates P from Q if(16) P ⊆ S,
Q ∩ S = ∅.
(See Figure 1C.7.) 1C.8. Prove that for each n > 1, every disjoint pair of sets P, Q in Π 0n can be e separated by a set in ∆ 0n . e Hint. To separate P from Q, reduce the pair X \ P, X \ Q. ⊣
1D. Parametrization and hierarchy theorems In the most general situation, a parametrization of a set S on I (with code set I ) is any surjection ð:I ։S on I onto S. Often we need parametrizations which are “nice”—e.g., we may want ð to be definable or to reflect some given structure on S.
1D.1]
1D. Parametrization and hierarchy theorems
27
P Q
S
Figure 1C.7. Separation. Here we are interested in the case when S is the restriction of a given pointclass Γ to some product space X , Γ ↾ X = {P ⊆ X : P ∈ Γ}. In fact we seek parametrizations of Γ ↾ X on product spaces. If P ⊆ Y × X and y ∈ Y, let Py be the y-section of P, Py = {x ∈ X : P(y, x)}, as in Figure 1D.1. A pointset G ⊆ Y × X is universal for Γ ↾ X , if G is in Γ and the map y 7→ Gy is a parametrization of Γ ↾ X on Y, i.e., for P ⊆ X ,(15) P ∈ Γ ⇐⇒ for some y ∈ Y, P = Gy . A pointclass Γ is Y-parametrized if for every product space X there is some G ⊆ Y ×X which is universal for Γ ↾ X . Let N0 , N1 , N2 , . . . be an enumeration of a basis for the topology of some product space X and define O ⊆ N × X by O(ε, x) ⇐⇒ (∃n)[x ∈ Nε(n) ]. Clearly O is open and each open set P ⊆ X is of the form S P = Oε = n Nε(n)
for some ε ∈ N , so that O is universal for Σ 01 ↾ X . Thus Σ 01 is N -parametrized and it e pointclassese Σ 0 and their duals Π 0 are is trivial to prove from this that all the Borel n n e e N -parametrized. The next theorem establishes a little more.
1D.1. The Parametrization Theorem for Σ 01 . For every perfect product space Y, e is Y-parametrized.(15)
Σ 01 e
Proof. Suppose N (Y, 0), N (Y, 1), . . . and N (X , 0), N (X , 1), . . . enumerate bases for the topology of Y and a fixed product space X respectively. Recall from Theorem 1A.2 that there is a function ó which assigns to each finite binary sequence u a
28
1. The basic classical notions
[1D.2
X P Py
y
Y
Figure 1D.1. The section above y. nbhd N Y, ó(u) in Y such that (i), (ii) and (iii) of 1A.2 hold. Using this ó, define G ⊆ Y × X by G(y, x) ⇐⇒ there exists a finite binary sequence u = (t0 , . . . , tn ) such that tn = 0, y ∈ N Y, ó(u) and x ∈ N (X , n).
It is immediate that G is open and hence every section Gy ⊆ X is open. The proof will be complete if we show that every open subset of X is a section of G, since then G will be universal for Σ 01 ↾ X and X was arbitrary. e there is a set of integers A such that If P ⊆ X is open, then x ∈ P ⇐⇒ (∃n)[n ∈ A & x ∈ N (X , n)].
Put
( 0 tn = 1
if n ∈ A, if n ∈ /A
and as in the proof of 1A.3 define the sequence {yn } in Y by yn = the center of N Y, ó(t0 , . . . , tn ) . The properties of ó imply that {yn } is Cauchy, so let y = limn→∞ yn . We claim that for this y, G(y, x) ⇐⇒ x ∈ P. If x ∈ P, then for some nwe have tn = 0 and x ∈ N (X , n), and by the properties of ó, y ∈ N Y, ó(t0 , . . . , tn ) , so by the definition of G we have G(y, x). Conversely, if G(y, x), then there is some u = (t0′ , t1′ , . . . , tn′ ) such that y ∈ N Y, ó(u) and tn′ = 0 and x ∈ N (X , n). Since y ∈ N Y, ó(t0 , t1 , . . . , tn ) , the sequences (t0 , . . . , tn ) and (t0′ , . . . , tn′ ) are compatible by the properties of ó. But binary sequences of the same length are compatible only when they are identical, so t0 = ⊣ t0′ , . . . , tn = tn′ = 0, hence tn = 0 and x ∈ N (X , n), so x ∈ P. 1D.2. Theorem. If a pointclass Γ is Y-parametrized, then so are the pointclasses ¬Γ and ∃Z Γ, where Z is any product space. In particular all the Borel pointclasses Σ 0n and e their duals Π 0n are Y-parametrized, where Y is any perfect product space. e
1E]
1E. The projective sets
29
Proof. If G ⊆ Y × X is universal for Γ ↾ X , then ¬G = Y × X \ G is obviously universal form ¬Γ ↾ X . Similarly, if G ⊆ Y × X × Z is universal for Γ ↾ (X × Z), define H ⊆ Y × X by H (y, x) ⇐⇒ (∃z)G(y, x, z) and verify immediately that H is universal for ∃Z Γ ↾ X . ⊣ The significance of parametrizations is evident in the next result which we formulate in a very general setting. 1D.3. The Hierarchy Lemma. Let Γ be a pointclass such that for every product space X and every pointset P ⊆ X × X in Γ, the diagonal P ′ = {x : P(x, x)} is also in Γ. If Γ is Y-parametrized, then some P ⊆ Y is in Γ but not in ¬Γ.(15) Proof. Let G ⊆ Y × Y be universal for Γ ↾ Y and take P = {y : G(y, y)}. By hypothesis P ∈ Γ. If ¬P ∈ Γ, then for some fixed y ∗ ∈ Y we would have G(y ∗ , y) ⇐⇒ ¬P(y) ⇐⇒ ¬G(y, y) which is absurd for y = y ∗ . ⊣ 1D.4. The Hierarchy Theorem for the Borel Pointclasses of Finite Order. If X is any perfect product space, then the following diagram of proper inclusions holds:
∆ 01 ↾ X e
( (
Σ 01 ↾ X e Π 01 ↾ X e
( (
∆ 02 ↾ X e
(
Σ 02 ↾ X e
···
···
(
Π 02 ↾ X · · · e
Diagram 1D.2. The Borel pointclasses of finite order.
Proof. We have the inclusions from 1B.1, so it is enough to prove that they are proper. From 1D.2 we know that Σ 0n and Π 0n are X -parametrized, hence by 1D.3 there is some P ⊆ X , P ∈ Σ 0n , P ∈ / Πe0n . Thuse∆ 0n ↾ X ( Σ 0n ↾ X and similarly ∆ 0n ↾ X ( Π 0n ↾ e e ↾ X , then e Σ 0 ↾ X would be closed e e ¬, X . On the other hand, if Σ 0ne↾ X = ∆ 0n+1 under n e e e 0 0 0 0 so Π n ↾ X ⊆ Σ n ↾ X contradicting P ∈ Σ n \ Π n . ⊣ e e e e
1E. The projective sets
We now introduce a second hierarchy of pointclasses by applying repeatedly the operations of negation and projection along N . For each pointclass Λ let ∃N Λ = {∃N P : P ∈ Λ} = {∃N P : P ∈ Λ ↾ (X × N ) for some X }. The Lusin pointclasses Σ 1n (n ≥ 1) are defined by the recursion e Σ 11 = ∃N Π 01 , e e Σ 1n+1 = ∃N ¬Σ 1n , e e
30
1. The basic classical notions
[1E.1
and as with the Borel pointclasses we define the dual and ambiguous Lusin pointclasses by(11,12) Π 1n = ¬Σ 1n , e1 e ∆ n = Σ 1n ∩ Π 1n . e e e Thus a pointset P ⊆ X is Σ 11 if there is a closed F ⊆ X × N such that for all x e P(x) ⇐⇒ (∃α)F (x, α),
P is Σ 12 ( if there is an open G ⊆ X × N × N such that e P(x) ⇐⇒ (∃α1 )(∀α2 )G(x, α1 , α2 ), etc. Similarly, P is Π 11 if there is an open G such that e P(x) ⇐⇒ (∀α)G(x, α),
P is Π 12 if there is a closed F such that e P(z) ⇐⇒ (∀α1 )(∃α2 )F (x, α1 , α2 ),
etc. The pointsets that occur in these Lusin pointclasses are the projective sets, the chief objects of our study. 1E.1. Theorem. The following diagram of inclusions holds among the Lusin pointclasses:
⊆ ∆ 11 e
⊆
Σ 11 e Π 11 e
⊆ ⊆
⊆ ∆ 12 e
Σ 12 e
···
···
⊆
Π 12 · · · e
Diagram 1E.1. The Lusin pointclasses. Proof. The inclusions Σ 1n ⊆ Π 1n+1 are proved by vacuous quantification, the same e in 1B.1. e way we showed Σ 0n ⊆ Π 0n+1 e e If F is a closed set, then F is Π 02 by 1B.1, so for some open G, e F (x) ⇐⇒ (∀t)G(x, t) ⇐⇒ (∀α)G x, α(0) .
Now the set G ′ ⊆ X × N defined by
G ′ (x, α) ⇐⇒ G x, α(0)
is also open since G is and the map
(x, α) 7→ x, α(0)
is continuous, hence F is Π 11 . Thus every closed set is Π 11 and then, by definition, e e every Σ 11 set is Σ 12 , from which e e 1 1 Σ n ⊆ Σ n+1 e e follows immediately by induction. The remaining inclusions in the diagram are trivial. ⊣
1E.2]
1E. The projective sets
31
To prove the closure properties of the Lusin pointclasses we need maps that allow us to code infinite sequences of irrational by single irrationals. Put (α)i = t 7→ α(hi, ti) , i.e.,
(α)i = â, where â(t) = α(hi, ti). There is a k-ary inverse of this function for each k ≥ 1, hα0 , . . . , αk−1 i(hi, ti) = αi (t) hα0 , . . . , αk−1 i(n) = 0
if i < k, if n 6= hi, ti for all t and i < k.
The maps (α, i) 7→ (α)i , (α0 , . . . , αk−1 ) 7→ hα0 , . . . , αk−1 i are obviously continuous and for each k and i < k, (hα0 , . . . , αk−1 i)i = αi . It is also useful to have a notation for the shift map, α ⋆ = (t 7→ α(t + 1)). Again, α 7→ α ⋆ is continuous. 1E.2. Theorem. Each Lusin pointclass Σ 1n is closed under continuous substitution, ∨, e & , ∃≤ , ∀≤ , ∀ù and ∃Y for every product space Y. 1 Each dual Lusin pointclass Π n is closed under continuous substitution, ∨, & , ∃≤ , ∀≤ , e ∃ù and ∀Y for every product space Y. Each ambiguous Lusin pointclass ∆ 1n is closed under ¬, ∨, & , ∃≤ , ∀≤ , ∃ù and ∀ù . e Borel order is ∆ 1 . In particular, every pointset of finite 1 e Proof. The results about Π 1n and ∆ 1n follow immediately from those about Σ 1n and e e of the closure properties of ∆ 1 . e the last assertion is a trivial consequence 1 1 0 e Closure of Σ 1 under continuous substitution follows from the closure of Π 1 under e e continuous substitution. ≤ ≤ Y 1 To prove closure of Σ 1 under ∨, & , ∃ , ∀ and ∃ we use quantifier contractions. For example, to prove eclosure under ∃N , assume that P(x, α) ⇐⇒ (∃â)F (x, α, â)
with F closed. Then
(∃α)P(x, α) ⇐⇒ (∃α)(∃â)F (x, α, â) ⇐⇒ (∃ã)F x, (ã)0 , (ã)1
and ∃N P is Σ 11 by closure of Π 01 under continuous substitution. e more example,esuppose To take one P(x, m) ⇐⇒ (∃â)F (x, m, â).
Then
(∀m ≤ n)P(x, m) ⇐⇒ (∀m ≤ n)(∃â)F (x, m, â) ⇐⇒ (∃ã)(∀m ≤ n)F x, m, (ã)m
and again ∀≤ P is Σ 11 by closure of Π 01 under continuous substitution and ∀≤ . e e
32
1. The basic classical notions
∆ 11 ↾ X e
( (
Σ 11 ↾ X e Π 11 ↾ X e
( (
∆ 12 ↾ X e
(
[1E.3 Σ 12 ↾ X e
···
···
(
Π 12 ↾ X · · · e
Diagram 1E.2. The Lusin pointclasses.
Closure of Σ 11 under ∃ù follows immediately from the equivalence e (∃t)(∃α)Q(x, t, α) ⇐⇒ (∃ã)Q x, ã(0), ã ⋆ . For every product space Y, there is a continuous surjection ð:N ։Y of N onto Y by 1A.1. Thus if P ⊆ X × Y, then
(∃y)P(x, y) ⇐⇒ (∃α)P x, ð(α)
and closure of Σ 11 under ∃Y follows from closure under continuous substitution e and ∃N . Finally, to prove closure of Σ 11 under ∀ù , suppose e P(x, t) ⇐⇒ (∃α)F (x, t, α) with F in Π 01 . Then e
(∀t)P(x, t) ⇐⇒ (∀t)(∃α)F (x, t, α)
⇐⇒ (∃ã)(∀t)F x, t, (ã)t ,
so ∀ù P is Σ 11 by closure of Π 01 under continuous substitution and ∀ù . e properties of e The closure Σ 1n for n > 1 follow by induction, using the same quantifier manipulations that we usedefor the case of Σ 11 . ⊣ e In addition to the obvious quantifier contractions (∃α)(∃â)P(α, â) ⇐⇒ (∃ã)P (ã)0 , (ã)1 , (∀α)(∀â)P(α, â) ⇐⇒ (∀ã)P (ã)0 , (ã)1 , we also used in this proof the equivalence
(∀t)(∃α)P(t, α) ⇐⇒ (∃ã)(∀t)P t, (ã)t .
This expresses the countable axiom of choice for pointsets. The dual equivalence (∃t)(∀α)P(t, α) ⇐⇒ (∀ã)(∃t)P t, (ã)t
looks a bit mysterious at first sight. We prove it by taking the negation of each side in the countable axiom of choice. Theorems 1D.1–1D.4 and 1E.1 yield immediately the following result. 1E.3. The Parametrization and Hierarchy Properties of the Lusin Pointclasses. For each n ≥ 1 and for each perfect product space Y, the pointclasses Σ 1n , Π 1n are Y-parametrized. Hence they satisfy the diagram 1E.2 of proper inclusions e (on e (12) the following page), where X is any perfect product space.
1F]
1F. Countable operations . . .
33
In the classical terminology the Σ 11 pointsets are called analytic or A-sets. They e include most of the sets one encounters in hard analysis. The Π 11 sets are coanalytic 1 e etc. or CA-sets, the Σ 2 sets are PCA-sets, the Π 12 sets are CPCA-sets, e e
Exercises
1E.4. If f : X → Y, let Graph(f) = {(x, y) : f(x) = y}. Prove that if f is continuous, then Graph(f) is closed. 1E.5. Prove that if f : X → Y is continuous and P is a Σ 1n subset of X , then e f[P] = {f(x) : P(x)} is Σ 1n . e 1E.6. Prove that for every pointset P ⊆ X ,
P is Σ 11 ⇐⇒ P = f[N ] for some continuous f, e P is Σ 1n+1 ⇐⇒ P = f[Q] for some Π 1n set Q ⊆ N and some continuous f. e e Hint. For the first assertion, suppose P is the projection of some closed subset C of X ×N . Consider C as a metric space with the metric it inherits from X ×N ; it is easily separable and complete, so by 1A.1, there is a continuous surjection f : N ։ C . Now P is the image of N under f followed by the continuous projection function. ⊣ We cannot replace N by an arbitrary perfect product space in this result, because of the next exercise. However, see 1G.12 for a related characterization of Σ 11 . e 1E.7. Prove that if f : R → X is continuous and F is a closed set of reals, the f[F ] is Σ 02 . e Hint. R is a countable union of compact sets. ⊣
Practically every specific pointset which comes up in the usual constructions of analysis and topology is easily shown to be projective—in fact, almost always, it is Σ 11 e or Π 11 . We only mention a couple of simple examples here, since we will meet several e interesting projective pointsets later on. 1E.8. On the space C [0, 1] of continuous real functions on the unit interval, put Q(f) ⇐⇒ f is differentiable on [0, 1], R(f) ⇐⇒ f is continuously differentiable on [0, 1], where at the endpoints we naturally take the one-sided derivatives. Prove that Q is Π 11 e and R is Σ 11 . e
1F. Countable operations and the transfinite Borel pointclasses
A countable pointset operation is any function Φ with domain some set of infinite sequences of pointsets and pointsets as values. We will often use the notation Φi Pi = Φ(P0 , P1 , P2 , . . . ).
34
1. The basic classical notions
[1F.1 Vù
The most W obvious countable operations are countable conjunction, , and countable V Wù disjunction, ù . Here ù i Pi and i Pi are defined when all the Pi are subsets of the same space X and V V x∈ ù i Pi ⇐⇒ i Pi (x) ⇐⇒ for all i ∈ ù, Pi (x), Wù W x ∈ i Pi ⇐⇒ i Pi (x) ⇐⇒ for some i ∈ ù, Pi (x). In set theoretic notation
Vù i
Pi =
T
i
Pi ,
Wù i
Pi =
S
i
Pi
whenever all the Pi are subsets of the same space. A pointclass Λ is closed under a countable operation Φ, if whenever P0 , P1 , . . . are all in Λ and Φi Pi is defined, then Φi Pi is also in Λ. 1F.1. Theorem. Let Γ be an N -parametrized pointclass which isWclosed under continù uous substitution. If Γ is closed underV∃ù , then it is closed under and if Γ is closed ù ù under ∀ , then it is also closed under . Proof. Suppose Pi ⊆ X , Pi ∈ Γ, let G ⊆ N ×X be universal and choose irrationals εi such that Pi = Gεi = {x ∈ X : G(εi , x)}. Now pick ε so that for every i, (ε)i = εi and set x ∈ P ⇐⇒ (∃i)G (ε)i , x . S Clearly P ∈ Γ by closure under continuous substitution and ∃ù and P = i Pi . ù The argument about ∀ is similar. ⊣ Wù Vù 0 0 1F.2. Corollary. Each Σ n is closed under each Π n is closed under and all W V e Σ 1n , Π 1n , ∆ 1n are closed undereboth ù and ù .(12) ⊣ e e e If Φ is a k-ary or countable set operation and Λ is a pointclass, put ΦΛ = {Φ(P0 , P1 , . . . ) : P0 , P1 , · · · ∈ Λ and Φ(P0 , P1 , . . . ) is defined}.
We have already used this notation in connection with ∃ù and ∃N . It is trivial to verify that if Λ is closed under continuous substitution, then W ∃ù Λ ⊆ ù Λ,
i.e., every projection along ù of a set in Λ can be written as a countable union of sets in Λ. This together with 1F.2 give us a new inductive characterization of the finite Borel pointclasses, Σ 01 = all open sets, e W Σ 0n+1 = ù ¬Σ 0n . e e Now the classWof all pointsets of finite Borel order is closed under ∃ù but it is not closed under ù ; for example, choose Gn ⊆ N to be in Σ 0n \ Π 0n and verify that e e S G = n {(n, α) : α ∈ Gn }
is not in any Σ 0n . This suggests an extension of the finite Borel hierarchy into the e transfinite as follows. Take Σ 01 = all open pointsets e
1F.3]
1F. Countable operations . . .
35
and for each ordinal number î > 1, let Wù S Σ 0î = ¬( ç 1, every pair of sets P, Q in Σ 0î is reducible by a pair e P ∗ , Q ∗ in Σ 0î .(16) e 1F.10. Prove that for each î > 1, every disjoint pair of sets P, Q in Π 0î can be e separated by a set in ∆ 0î .(16) e
1G. Borel functions and isomorphisms Let Λ be a fixed pointclass and let
f:X →Y be a function. We say that f is Λ-measurable, if for each basic nbhd Ns ⊆ Y, the inverse image f −1 [Ns ] is in Λ. This notion is due to Lebesgue.(10) Here we are mostly interested in Borel measurable or simply Borel functions. A Borel isomorphism between two spaces is a bijection f:X →Y such that both f and its inverse are Borel measurable. The main result of this section is that every perfect product space is both Borel isomorphic with N and the continuous one-to-one image of some closed subset of N . We will also show that the Lusin pointclasses are closed under Borel substitution. Thus in studying projective sets we can often simplify proofs by assuming that the space under consideration is N . We will leave for the exercises some very interesting results about Σ 0î -measurable e functions. Let us first dispose of the easy result. 1G.1. Theorem. If f : X → Y is a Borel function and P ⊆ Y is in any of the pointclasses B, ∆ 1n , Σ 1n , Π 1n , then f −1 [P] is in the same pointclass. e e In particular, ethe collection of Borel functions is closed under composition.
38
1. The basic classical notions
[1G.2
Proof. A simple induction on î shows that if f is Borel and P is Σ 0î , then f −1 [P] e Y, g : Y → Z is Borel. Thus B is closed under Borel substitution. Also, if f : X → are both Borel and h : X → Z is the composition, h(x) = g f(x) , then for each open set P ⊆ Z,
h −1 [P] = f −1 g −1 [P] ,
so h −1 [P] is Borel and h is Borel measurable. For the rest, notice that V f(x) = y ⇐⇒ s [y ∈ Ns =⇒ f(x) ∈ Ns ],
so that the graph of f
Graph(f) = {(x, y) : f(x) = y} is Borel. Now for any P ⊆ Y,
P f(x) ⇐⇒ (∃y)[P(y) & f(x) = y]
⇐⇒ (∀y)[P(y) ∨ f(x) 6= y].
These equivalences, the fact that B ⊆ ∆ 11 and the closure properties of the pointclasses ∆ 1n , Σ 1n , Π 1n imply immediately that if eP is in one of them, then so is f −1 [P]. ⊣ e e e We now go to the transfer theorems which often allow us to study just subsets of N instead of arbitrary pointsets. The first of these is a more refined statement of Theorem 1A.1.(18) 1G.2. Theorem. For every product space X there is a continuous surjection ð:N ։X and a closed set A ⊆ N such that ð is one-to-one on A and ð[A] = X . Moreover, there is a Borel injection f:X N which is precisely the inverse of ð restricted to A, i.e., for all α ∈ A, f ð(α) = α and for all x ∈ X , f(x) ∈ A and ð f(x) = x. Proof. To begin with, let ñ:N ։X be the surjection defined in the proof of 1A.1 and for x ∈ X , put g(x) = α, where α(n) = least k such that d (x, rk ) ≤ 2−n−2 . It is very simple to check that for all x ∈ X , ñ g(x) = x, so g is an injection. Moreover, if we put B = g[X ], then g is precisely the inverse of ñ restricted to B, since α ∈ B =⇒ α = g(x) for some x, =⇒ g ñ(α) = g ñ g(x) = g(x) = α.
1G.2]
1G. Borel functions and isomorphisms
39
If g(x) = α, then a(n) = k ⇐⇒ d (x, rk ) ≤ 2−n−2 & (∀s < k)[d (x, rs ) > 2−n−2 ]. Thus if Bn,k = {α : α(n) = k}, each g [Bn,k ] is a Borel subset of X . It follows that for each basic nbhd N = {α : α(0) = k0 , . . . , α(n − 1) = kn−1 } in N , the set −1
g −1 [N ] = g −1 [B0,k0 ] ∩ · · · ∩ g −1 [Bn−1,kn−1 ] is Borel and g is a Borel function. Now, easily h α ∈ B ⇐⇒ (∀n) d ñ(α), rα(n) ≤ 2−n−2
i & ∀k < α(n) d ñ(α), rk > 2−n−2 ,
so B is a Π 02 subset of N . We must refine the construction a bit to get ð and A with e the same properties, with A a closed set. Put B in normal form α ∈ B ⇐⇒ (∀n)(∃s)R(α, n, s),
where R is a clopen pointset by 1B.7 and define A ⊆ N × N by (α, â) ∈ A ⇐⇒ (∀n) R α, n, â(n) & ∀k < â(n) ¬R(α, n, k) .
Clearly A is closed. Moreover, the projection ó : N × N → N , ó(α, â) = α takes A onto B and is one-to-one on A, since (α, â) ∈ A =⇒ â(n) = least k such that R(α, n, k). Hence the composition ð = ñ ◦ ó takes A onto X and is continuous, one-to-one. It is trivial to check that the inverse of ð f(x) = g(x), n 7→ least k such that R g(x), n, k
is Borel. The proof is completed by carrying A to N via some trivial homeomorphism of N with N × N , e.g., the map n0 , n1 , n2 , . . . 7→ (n0 , n2 , n4 , . . . ), (n1 , n3 , n5 , . . . ) . ⊣
The function f of this proof is an example of an interesting class of functions. Let us temporarily call a function f:X Y a good Borel injection if (1) f is a Borel injection, (2) there is a Borel surjection g:Y ։X such that g ◦ f is the identity on X , i.e., g f(x) = x (x ∈ X ).
We refer to any such g as a Borel inverse of f. It will turn out that every Borel injection is a good Borel injection. This is a special case of a fairly difficult theorem which we will prove in 2E and again in Chapter 4. Here we only need show that enough good Borel injections exist.
40
1. The basic classical notions
[1G.3
Notice that if f : X Y is a good Borel injection, then y ∈ f[X ] ⇐⇒ f g(y) = y
with g any Borel inverse of f, so f[X ] is a Borel set. Moreover, if P is any Borel subset of X , then y ∈ f[P] ⇐⇒ y ∈ f[X ] & g(y) ∈ P, so that f[P] is Borel. Thus the image of a Borel set by a good Borel injection is Borel. It is also immediate that the class of good Borel injections is closed under composition. 1G.3. Lemma. For every perfect product space X , there are good Borel injections f : X N, h : N X. Proof. We have already constructed f in 1G.2. To construct h, define first h1 : N C by h1 (α) = â, where â(n) =
(
0 if α (n)0 = (n)1 , 1 if α (n)0 = 6 (n)1 .
It is trivial to verify that h1 is a Borel function, and h i â ∈ h1 [N ] ⇐⇒ (∀n) â(n) = â h(n)0 , (n)1 i h i & (∀n)(∀k) [â(n) = 0 & â(k) = 0 & (n)0 = (k)0 ] =⇒ (n)1 = (k)1
& (∀n)(∃k)[â(hn, ki) = 0],
so that h1 [N ] is Borel. Define now g1 : C ։ N by ( the constant function 0 if â ∈ / h1 [N ], g1 (â) = α if â ∈ h1 [N ], where α(n) = the unique m such that â(hn, mi) = 0 and verify easily that g1 is a Borel inverse of h1 , so that h1 is a good Borel injection. Now let ð:CX be the continuous injection constructed in 1A.3 with M = X . Since C is compact and ð is a continuous injection, we know that ð[C] is compact; in any case, we can compute ð[C] using the function ó of 1A.2, V W x ∈ ð[C] ⇐⇒ n u [u = (t0 , . . . , tn−1 ) for some t0 , . . . , tn−1 & x ∈ Nó(u) ]. For an inverse to ð, take ( the constant 0 function ñ(x) = the unique α ∈ C such that ð(α) = x
If B = {α : α(0) = k0 , . . . , α(n) = kn }
if x ∈ / ð[C], if x ∈ ð[C].
1G.4]
1G. Borel functions and isomorphisms
41
h R N0
X0
f[X0 ]
h[N0 ]
N1 N1
X1
I
N0 = N Nn+1 = fh[Nn ]
f
X0 = X Xn+1 = hf[Xn ]
Diagram 1G.1. is a typical nbhd in C, then ñ(x) ∈ B ⇐⇒ ñ(x)(0) = k0 & · · · & ñ(x)(n) = kn , so to prove the ñ is Borel it is enough to show that for each n, the relation Pn (x) ⇐⇒ ñ(x)(n) = 0 is Borel. This is true, since Pn (x) ⇐⇒ x ∈ / ð[C] ∨
W
u [u
= (t0 , . . . , tn−1 ) for some t0 , . . . , tn−1 & tn−1 = 0 & x ∈ Nó(u) ].
Now h = ð ◦ h1 is a good Borel injection of N into X .
⊣ (18)
1G.4. Theorem. Every perfect product space is Borel isomorphic with N . ¨ Proof. Recall the classical Schroeder-Bernstein Theorem, whose proof constructs from given injections h : N X and f : X N a bijection g : N → X . We will verify that if h, f are good Borel injections, then the resulting bijection is a Borel isomorphism. Define the sequences of sets N0 , N1 , . . . , X0 , X1 , . . . recursively by the equations N0 = N X0 = X Nn+1 = fh[Nn ] Xn+1 = hf[Xn ], see Diagram 1G.1. An easy induction shows that Nn ⊇ f[Xn ] ⊇ Nn+1 , Xn ⊇ h[Nn ] ⊇ Xn+1 , so that N = N0 ⊇ f[X0 ] ⊇ N1 ⊇ f[X1 ] ⊇ N2 ⊇ f[X2 ] ⊇ · · · , X = X0 ⊇ f[N0 ] ⊇ X1 ⊇ f[N1 ] ⊇ X2 ⊇ f[N2 ] ⊇ · · · .
42
1. The basic classical notions
Put also N∗ = and notice that X∗ = and since h is an injection,
T
n
T
n
Xn ⊇
Nn , T
n
X∗ =
h[Nn ] ⊇
T
n
T
n
[1G.5
Xn
Xn+1 = X ∗ ,
T T h[N ∗ ] = h[ n Nn ] = n h[Nn ] = X ∗ .
Thus h gives a bijection on N ∗ with X ∗ . On the other hand,
N = (N0 − f[X0 ]) ∪ (f[X0 ] − N1 ) ∪ (N1 − f[X1 ]) ∪ (f[X1 ] − N2 ) ∪ · · · ∪ N ∗ * k 3 k + s s X = (X0 − h[N0 ]) ∪ (h[N0 ] − X1 ) ∪ (X1 − h[N1 ]) ∪ (h[N1 ] − X2 ) ∪ · · · ∪ X ∗ where the sets in these unions are disjoint. Moreover, h is a bijection of Nn \ f[Xn ] with h[Nn ] \ Xn+1 , since h is an injection and f[Xn ] ⊆ Nn , so that h Nn \ f[Xn ] = h[Nn ] \ hf[Xn ] = h[Nn ] \ Xn+1 ,
and similarly, f is a bijection of Xn \ h[Xn ] with f[Xn ] \ Nn+1 . So we have a bijection of N with X , ( h(α) if α ∈ N ∗ or α ∈ Nn \ f[Xn ] for some n, g(α) = −1 f (α) if α ∈ / N ∗ and α ∈ f[Xn ] \ Nn+1 for some n. It remains to verify that g is Borel. Recall that good Borel injections map Borel sets onto Borel sets. This implies that all the sets Nn , Xn are Borel, hence N ∗ , X ∗ and all the differences Nn \ f[Xn ], f[Xn ] \ Nn+1 are Borel. From this it follows immediately that g is Borel. ⊣
Exercises Let us start with a very simple representation of Borel sets which comes out of 1G.2. 1G.5. Prove that every Borel set is the continuous, injective image of a closed set of irrationals; i.e., if P ⊆ X is Borel, then there exists a continuous ð : N → X and a closed B ⊆ N such that ð is one-to-one on B and ð[B] = P. Hint. Let C be the class of all P ⊆ X which are continuous, injective images of some closed B ⊆ N . Every closed P is in C: just let ð : N → X , let A be as in 1G.2 and take B = ð−1 [P] ∩ A. If P is open, then the same B is the intersection of a closed and an open set, which makes it Π 02 ; we can now use the trick in the proof of 1G.2 to replace it by e a closed set. For each finite sequence k0 , . . . , kn−1 , let N (k0 , . . . , kn−1 ) = {α : α(0) = k0 , . . . , α(n − 1) = kn−1 }.
Each N (k0 , . . . , k Sn−1 ) is trivially homeomorphic with N . Suppose P = n Pn , each Pn ∈ C, and n 6= m =⇒ Pn ∩ Pm = ∅.
1G.8]
1G. Borel functions and isomorphisms
43
We may assume then that there are closed sets Bn ⊆ N (n) and continuous maps ðn : N (n) → X such that ðn [Bn ] = Pn and ðn is injective on Bn . Take B = {α : α ⋆ ∈ Bα(0) }, with α ⋆ = t 7→ α(t + 1) and ð(α) = ðα(0) (α ⋆ ). T Suppose P = n Pn with Bn , ðn again as above. Let (α)i be defined as in 1E and put α ∈ B ⇐⇒ (∀n)[(α)n ∈ Bn ] & (∀n)(∀m) ðn (α)n = ðm (α)m & (∀t)[t 6= h(t)0 , (t)1 i =⇒ α(t) = 0].
Clearly B is closed. Let
ð(α) = ð0 (α)0 T and verify that ð is one-to-one on B and ð[B] = n Pn . Now let D be the class of all P ⊆ X such that both P and X \ P are in C. We have shown that D containsTthe open sets, and it is certainly closed under complementation. If each Pn ∈ D, then n Pn ∈ C, as above, and if we let Qn = X \ Pn , then T S S S X \ n Pn = n Qn = n Qn \ i i, ϕi (yn ) = îi , so by the basic property of the semiscale we have x ∈ P. (iii)=⇒(iv) is trivial. (iv)=⇒(i). Assume P = Auκ Pu with each Pu closed and put V C (x, f) ⇐⇒ n [x ∈ Pf↾n ]. Clearly C is closed and
x ∈ pC ⇐⇒ (∃f)C (x, f) ⇐⇒ (∃f)(∀n)[x ∈ Pf↾n ] ⇐⇒ x ∈ P.
⊣
Suslin’s original definition of analytic sets was via the operation A , A = A ℵ0 and the essential content of the equivalences (i) ⇐⇒ (iii) ⇐⇒ (iv) was already announced in the basic papers Suslin [1917], Lusin [1917]. Let S(κ) = Sκ be the pointclass of all κ-Suslin sets, so in particular S(ℵ0 ) = Σ 11 . e 2B.2. Theorem. For each cardinal κ ≥ ℵ0 , the pointclass Sκ is closed Vù under Borel Y substitution, ∃ for every product space Y, countable conjunction, , disjunction of W length κ, κ , and the operation A κ . Moreover, if ë ≤ κ, then Së ⊆ Sκ ; in particular every Σ 11 pointset is κ-Suslin. e Proof. Closure of Sκ under continuous substitution is immediate, so we can use it in the arguments below. To prove closure under ∃N , suppose C ⊆ X × N × ù κ is closed and P(x, α) ⇐⇒ (∃f)C (x, α, f), so that (∃α)P(x, α) ⇐⇒ (∃α)(∃f)C (x, α, f). Let
ð(î) = ð1 (î), ð2 (î) be a bijection of κ with ù × κ as in the proof of 2B.1 and notice that the mapping ñ(g) = (g1 , g2 ) where
g1 (n) = ð1 g(n) , g2 (n) = ð2 g(n) ,
is a homeomorphism of ù κ with ù ù × ù κ = N × ù κ. Thus if we define C ∗ (x, g) ⇐⇒ C (x, g1 , g2 ),
2B.2]
2B. κ-Suslin sets
55
the set C ∗ is closed in X × ù κ and (∃α)P(x, a) ⇐⇒ (∃g)C ∗ (x, g), so ∃N P is κ-Suslin. We can now prove closure of Sκ under ∃Y using closure under ∃N and the fact that every Y is a continuous image of N . If Pî = pCî for each î < κ, put C (x, f) ⇐⇒ Cf(0) (x, f ⋆ ), where by definition f ⋆ (n) = f(n + 1). Clearly C is closed and W
î îj . Easily S is a tree on ù × ℵ1 . The claim is that P(α) ⇐⇒ S(α) is not wellfounded. Notice that for any fixed α and u = (s0 , . . . , sk−1 ),
u is T -compatible with α(0), . . . , α(n − 1)
⇐⇒ k ≤ n & α(0), s0 , . . . , α(k − 1), sk−1 ∈ T ⇐⇒ k ≤ n & u ∈ T (α).
Using the condition length(un ) ≤ n we then have (î0 , . . . , în−1 ) ∈ S(α) ⇐⇒
α(0), î0 , . . . , α(n − 1), în−1 ∈ S
⇐⇒ for every i, j < n, if ui , uj are in T (α) and ui is an initial segment of uj , then îi > îj . This observation implies immediately that if (î0 , î1 , . . . ) is an infinite branch of S(α), then the mapping ui 7→ îi is a rank function on T (α), so that T (α) is wellfounded. Conversely, if T (α) is wellfounded, let ñ be a rank function on T (α), put îi = ñ(ui ) and check immediately that (î0 , î1 , . . . ) is an infinite branch of S(α), so that S(α) is not wellfounded. ⊣ We will prove later much better representation theorems for Π 11 and Σ 12 along these e ℵ1 elements e lines. However this result already implies that a Σ 12 set with more than e has a perfect subset.
2E.1]
2E. The Suslin Theorem
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C A X B
Figure 2E.1. Separation.
2E. The Suslin Theorem Fix a space X and an ordinal ë > ù. A collection C of subsets of X is a ë-algebra if ∅ ∈ C and C is closed under complementation and unions of length less than ë, i.e., î < ë and for all ç < î, Aç ∈ C =⇒
S
ç ù. Let Bë′ be the collection of all ë-Borel pointsets P ⊆ Y, such that for every Borel function f : X → Y, f −1 [P] is ë-Borel. Clearly Bë′ contains all open sets and is closed under ¬ and unions of length less than ë, hence Bë′ = Bë and Bë is closed under Borel substitution. We will leave for the exercises the remaining easy closure properties of Bë . Here we want to concentrate on the Strong Separation Theorem and its corollary, the Suslin Theorem which is the chief construction principle of classical descriptive set theory. Recall from 1C that a set C separates A from B if A ⊆ C , B ∩ C = ∅ (see Figure 2E.1). 2E.1. The Strong Separation Theorem (Lusin). Suppose κ is an infinite cardinal and A, B are disjoint κ-Suslin subsets of some perfect product space X . There exists a (κ + 1)-Borel set C which separates A from B.(8) Proof. We may assume that A, B are subsets of N , since X is Borel isomorphic with N and both Sκ and Bκ+1 are closed under Borel substitution. The key to the proof is the following simple combinatorial fact about separating sets. Suppose S S A = i∈I Ai , B = j∈J Bj
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2. κ-Suslin and ë-Borel
[2E.1
are unions of sets, where the index sets I , J are quite arbitrary, suppose that for each i ∈ I , j ∈ J there is a set Ci,j which separates Ai from Bj . Then the set S T C = i∈I j∈J Ci,j
separates T A from B. To prove S this, notice S Tthat for each i, j, Ai ⊆ Ci,j , hence Ai ⊆ j∈J Ci,j , hence A = i∈I Ai ⊆ i∈I j∈J Ci,j = C . On the other hand, for S S each i, j, Bj ⊆ N \ Ci,j , hence B = j∈J Bj ⊆ j∈J (N \ Ci,j ) and since this holds for arbitrary i, T S T T B ⊆ i∈I j∈J (N \ Ci,j ) = i∈I (N \ j∈J Ci,j ) S T = N \ i∈I j∈J Ci,j = N \ C.
Suppose now that A and B are disjoint κ-Suslin sets of irrationals, so there are trees T and S on ù × κ and A = p[T ], B = p[S]. We give two proofs of the result—first a simple argument by contradiction and then a constructive proof which actually exhibits a (κ + 1)-Borel set C that separates A from B. Proof by contradiction. Assume that A cannot be separated from B by a (κ+1)-Borel set. Since S A = p[T ] = t∈ù,î
